A Teaching Note on Simes’ Test and the BH Procedure

Abstract

In many traditional graduate courses in mathematical statistics in China, the main emphasis is still on point estimation, interval estimation and classical single-hypothesis testing, while topics such as global testing and multiple hypothesis testing are often treated very briefly or omitted altogether. At the same time, global and multiple testing have become standard tools in modern applications such as genomics, brain imaging and large-scale A/B testing, where hundreds or thousands of hypotheses are tested simultaneously. This teaching note is designed to help bridge that gap. We present a self-contained proof of Simes’ test based on order statistics. We then explain how Simes’ test is related to the Benjamini–Hochberg (BH) procedure for controlling the false discovery rate (FDR) in multiple hypothesis testing. R implementations and simulations that can be reproduced by students are provided. The aim is pedagogical rather than innovative: to make global testing and multiple testing accessible in a graduate course.